For more than 140 years the chemical composition of our Sun has been considered typical of solar-type stars. Our highly differential elemental abundance analysis of unprecedented accuracy (~0.01 dex) of the Sun relative to solar twins, shows that the Sun has a peculiar chemical composition with a ≈20% depletion of refractory elements relative to the volatile elements in comparison with solar twins. The abundance differences correlate strongly with the condensation temperatures of the elements. A similar study of solar analogs from planet surveys shows that this peculiarity also holds in comparisons with solar analogs known to have close-in giant planets while the majority of solar analogs without detected giant planets show the solar abundance pattern. The peculiarities in the solar chemical composition can be explained as signatures of the formation of terrestrial planets like our own Earth.